Terminating decimals

[valleeny]

How do we handle terminating decimals question like the one below?

If r and s are positive integers and the ratio r/s is expressed as a decimal, is r/s a terminating decimal?
(1) 90<r<100
(2) s=4

Now I read the explanation for the answer and it says any number divided by 4 is terminating decimal. How can you draw qucik conclusions for all the other divisors?

For eg.
Any number divided by :
1, has a terminating decimal
2, has a terminating decimal (0.5, 1)
3, does not have a terminating decimal (0.333..., 0.666..., 1)
4, has a terminating decimal (0.25, 0.5, 1)
5, has a terminating decimal (0.2, 0.4, 0.6, 0.8, 1)
6, may or may not have a terminating decimal (0.166..., 0.333..., 0.5, 0.666..., 0.833...)

Is this the right way to evaluate each divisor whether they will yield a terminating decimal? Or is there a set of rules that will be easier memorized than derived on the spot?

[cameronwu]

As far as I know, there's no other way around it other than being familiar with property of numbers. The building blocks of division don't get much more basic than this, unfortunately.

[Testluv]

Is this the right way to evaluate each divisor whether they will yield a terminating decimal? Or is there a set of rules that will be easier memorized than derived on the spot?

If you have the fraction completely simplified, and in the prime factorization of the denominator you only have 2s or you only have 5s (or only 2s and 5s), then it is a terminating decimal. If you have the fraction completley simplified, and in the prime factorization of the denominator, there are primes other than 2 and/or 5, then it is a repeating (non-terminating) decimal.

Now I read the explanation for the answer and it says any number divided by 4 is terminating decimal.

Yes, because 4 has only 2s in its prime factorization. Therefore, any number dividied by 4 is a terminating decimal. The same can be said about any number divided by 8, and so on.

[valleeny]

Thanks Testluv. That was a good shortcut to know. I think it's possible but I don't suppose it's necessary to understand the basics behind it. I'm just gonna remember this rule.